Abstract: Accurate experimental values for the vibrational ground tone or fundamental
vibrational energy splitting of H$_2$, HD, and D$_2$ are presented. Absolute
accuracies of $2\times10^{-4}$ cm$^{-1}$ are obtained from Doppler-free laser
spectroscopy applied in a collisionless environment. The vibrational splitting
frequencies are derived from the combination difference between separate
electronic excitations from the $X^{1}\Sigma_{g}^{+}, v=0, J$ and $v=1, J$
vibrational states to a common $EF^{1}\Sigma^{+}_{g}, v=0, J$ state. The
present work on rotational quantum states $J=1,2$ extends the results reported
by Dickenson et al. on $J=0$ [Phys. Rev. Lett. 110 (2013) 193601]. The
experimental procedures leading to this high accuracy are discussed in detail.
A comparison is made with full \emph{ab initio} calculations encompassing
Born-Oppenheimer energies, adiabatic and non-adiabatic corrections, as well as
relativistic corrections and QED-contributions. The present agreement between
the experimental results and the calculations provides a stringent test on the
application of quantum electrodynamics in molecules. Furthermore, the combined
experimental-theoretical uncertainty can be interpreted to provide bounds to
new interactions beyond the Standard Model of Physics or fifth forces between
hadrons.